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One of the many famous theorems proven by William Thurston is that a closed connected smooth manifold $M$ admits a codimension 1 smooth foliation if and only if $\chi(M)=0$:

W.P. Thurston, Existence of codimension-one foliations. Ann. of Math. (2) 104 (1976), no. 2, 249–268.

The "only if" part of this theorem, of course, is much simpler and older.

My question is: What happens in the topological category?

I convinced myself (although I did not write down a detailed proof) that if a closed connected topological manifold $M$ admits a topological codimension 1 foliation, then $\chi(M)=0$. (The proof imitates the "standard" smooth argument, using a map from $M$ to the symmetric product of $M$ with itself, instead of constructing a line bundle on $M$ in the smooth case.) The true question is if the hard part of Thurston's theorem holds in the topological category:

Question. Suppose that $M$ is a closed connected topological manifold with $\chi(M)=0$. Does $M$ admit a codimension 1 foliation?

I could not find anything on MathSciNet using the obvious algorithm.

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  • $\begingroup$ I will wear the down-vote on this question as a badge of honor. $\endgroup$ – Moishe Kohan Apr 5 at 20:46
  • $\begingroup$ Hello Moishe. Thurston's construction has two steps: 1) build the foliation on the complement of "holes"; 2) extend it through the holes. Since the holes are contained in a small neighborhood of a finite union of loops, you can smoothen the manifold there, hence the step 2 would make no problem. Today, the step 1 falls to the general methods of Gromov's h-principle, e.g. the Eliashberg-Mishachev Holonomic Approximation Theorem; so the actual question is to extend these methods to the topological case. Good luck. $\endgroup$ – Gael Meigniez Apr 14 at 6:48
  • $\begingroup$ Maybe a more interesting and more accessible question would be to build codimension-one foliations in the PL frame. $\endgroup$ – Gael Meigniez Apr 14 at 6:50

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